Below are listed rank-two temperaments supported by the 31edo patent val, below the indicated cutoff in TE badness.

5-limit temperaments with badness below 0.1

Listed is the wedgie and the TE badness times 1000 for four temperaments with badness less than 0.1.

1	<<1 4 4]]	Meantone	7.381	81/80
2	<<15 -2 -38]]	Luna	20.576	274877906944/274658203125
3	<<8 1 -17]]	Würschmidt	40.603	393216/390625
4	<<7 -3 -21]]	Orson	40.807	2109375/2097152

7-limit temperaments with badness below 0.06

Listed is the wedgie and the TE badness times 1000 for 19 temperaments with badness less than 0.06.

Rank	Wedgie	Name	Badness	Commas
1	<<22 -5 3 -59 -57 21]]	Tertiaseptal	12.995	2401/2400 65625/65536
2	<<1 4 10 4 13 12]]	Meantone	13.707	81/80 126/125
3	<<6 -7 -2 -25 -20 15]]	Miracle	16.742	225/224 1029/1024
4	<<16 2 5 -34 -37 6]]	Hemiwürschmidt	20.307	2401/2400 3136/3125
5	<<7 -3 8 -21 -7 27]]	Orwell	20.735	225/224 1728/1715
6	<<10 9 7 -9 -17 -9]]	Myna	27.044	126/125 1728/1715
7	<<9 5 -3 -13 -30 -21]]	Valentine	31.056	126/125 1029/1024
8	<<38 -3 8 -93 -94 27]]	Quasiorwell	35.832	2401/2400 29360128/29296875
9	<<3 12 -1 12 -10 -36]]	Mothra/Cynder	37.146	81/80 1029/1024
10	<<15 -2 -5 -38 -50 -6]]	Hemithirds	44.284	1029/1024 3136/3125
11	<<60 -8 11 -152 -151 48]]	Subneutral	45.792	2401/2400 274877906944/274658203125
12	<<4 16 9 16 3 -24]]	Squares	45.993	81/80 2401/2400
13	<<5 -11 -12 -29 -33 3]]	Tritonic	47.578	225/224 50421/50000
14	<<11 13 17 -5 -4 3]]	Nusecond	50.389	126/125 2430/2401
15	<<17 6 15 -30 -24 18]]	Semisept	50.472	1728/1715 3136/3125
16	<<8 1 18 -17 6 39]]	Würschmidt	50.776	225/224 8748/8575
17	<<23 -1 13 -55 -44 33]]	Grendel	51.834	6144/6125 16875/16807
18	<<2 8 -11 8 -23 -48]]	Mohajira	55.714	81/80 6144/6125
19	<<13 -10 6 -46 -27 42]]	Slender	56.934	225/224 589824/588245

11-limit temperaments with badness below 0.05

Listed is the wedgie and the TE badness times 1000 for 68 temperaments with badness less than 0.05.

Rank	Wedgie	Name	Badness	Commas
1	<<6 -7 -2 15 -25 -20 3 15 59 49]]	Miracle	10.684	225/224 243/242 441/440
2	<<7 -3 8 2 -21 -7 -21 27 15 -22]]	Orwell	15.231	99/98 121/120 176/175
3	<<44 -10 6 79 -118 -114 -27 42 218 201]]	Hemitert	15.633	2401/2400 3025/3024 65625/65536
4	<<9 5 -3 7 -13 -30 -20 -21 -1 30]]	Valentine	16.687	121/120 126/125 176/175
5	<<10 9 7 25 -9 -17 5 -9 27 46]]	Myna	16.842	126/125 176/175 243/242
6	<<1 4 10 18 4 13 25 12 28 16]]	Meantone	17.027	81/80 99/98 126/125
7	<<38 -3 8 64 -93 -94 -30 27 159 152]]	Quasiorwell	17.540	2401/2400 3025/3024 5632/5625
8	<<15 -2 -5 22 -38 -50 -17 -6 58 79]]	Hemithirds	19.003	385/384 441/440 3136/3125
9	<<23 -1 13 42 -55 -44 -13 33 101 73]]	Grendel	19.845	540/539 1375/1372 5632/5625
10	<<16 2 5 40 -34 -37 8 6 86 95]]	Hemiwürschmidt	21.069	243/242 441/440 3136/3125
11	<<1 4 10 -13 4 13 -24 12 -44 -71]]	Meanpop	21.543	81/80 126/125 540/539
12	<<4 16 9 10 16 3 2 -24 -32 -3]]	Squares	21.636	81/80 99/98 121/120
13	<<17 6 15 27 -30 -24 -16 18 42 24]]	Semisept	22.476	176/175 540/539 1331/1323
14	<<5 -11 -12 -3 -29 -33 -22 3 31 33]]	Tritonic	23.659	121/120 225/224 441/440
15	<<8 1 18 20 -17 6 4 39 43 -6]]	Würschmidt	24.413	99/98 176/175 243/242
16	<<13 -10 6 17 -46 -27 -18 42 74 27]]	Slender	25.342	225/224 385/384 1331/1323
17	<<2 8 20 5 8 26 1 24 -16 -55]]	Migration	25.516	81/80 121/120 126/125
18	<<11 13 17 12 -5 -4 -19 3 -17 -25]]	Nusecond	25.621	99/98 121/120 126/125
19	<<3 12 -1 -8 12 -10 -23 -36 -60 -19]]	Mothra	25.642	81/80 99/98 385/384
20	<<2 8 -11 5 8 -23 1 -48 -16 52]]	Mohajira	26.064	81/80 121/120 176/175
21	<<29 -8 11 57 -80 -64 -10 48 160 122]]	Eris	27.621	540/539 1375/1372 65625/65536
22	<<16 2 5 9 -34 -37 -41 6 14 8]]	Hemiwur	29.270	121/120 176/175 1375/1372
23	<<21 -9 -7 37 -63 -70 -14 9 117 128]]	Triwell	29.807	385/384 441/440 456533/455625
24	<<22 -5 3 24 -59 -57 -38 21 73 57]]	Tertia	30.171	385/384 1331/1323 1375/1372
25	<<82 -13 14 143 -211 -208 -57 69 377 353]]		30.609	2401/2400 3025/3024 369140625/369098752
26	<<3 12 -1 23 12 -10 26 -36 12 68]]		31.334	81/80 540/539 1029/1024
27	<<7 -3 8 33 -21 -7 28 27 87 65]]		31.438	225/224 441/440 1728/1715
28	<<67 -11 19 121 -173 -158 -40 75 319 274]]		32.121	3025/3024 180224/180075 703125/702464
29	<<6 -7 -2 -16 -25 -20 -46 15 -13 -38]]		32.946	99/98 176/175 1029/1024
30	<<10 9 7 -6 -9 -17 -44 -9 -45 -41]]		33.434	99/98 126/125 385/384
31	<<8 1 -13 20 -17 -43 4 -33 43 101]]		33.436	126/125 243/242 385/384
32	<<32 4 10 49 -68 -74 -33 12 100 103]]		34.814	2401/2400 3025/3024 3136/3125
33	<<22 -5 3 55 -59 -57 11 21 145 144]]		35.576	243/242 441/440 703125/702464
34	<<28 -12 1 39 -84 -77 -35 36 132 106]]		36.493	385/384 1375/1372 14641/14580
35	<<14 -6 16 35 -42 -14 7 54 102 43]]		38.377	225/224 243/242 2420/2401
36	<<25 7 2 47 -47 -67 -12 -15 85 125]]		39.162	441/440 3388/3375 6144/6125
37	<<4 -15 9 10 -33 3 2 63 75 -3]]		39.595	99/98 243/242 385/384
38	<<31 0 0 62 -72 -87 -9 0 144 174]]		39.9210	441/440 3136/3125 41503/41472
39	<<3 -19 -1 -8 -37 -10 -23 51 47 -19]]		40.217	99/98 121/120 1029/1024
40	<<19 14 4 32 -22 -47 -15 -30 26 76]]		40.539	126/125 176/175 14641/14580
41	<<54 -1 13 104 -127 -131 -22 33 245 247]]		41.496	2401/2400 5632/5625 46656/46585
42	<<8 1 -13 -11 -17 -43 -45 -33 -29 14]]		41.680	126/125 176/175 14641/14406
43	<<9 5 -3 38 -13 -30 29 -21 71 117]]		42.204	126/125 540/539 1029/1024
44	<<12 -14 -4 -1 -50 -40 -43 30 46 11]]		42.687	121/120 225/224 1029/1024
45	<<12 17 27 30 -1 9 6 15 11 -9]]		42.719	99/98 126/125 243/242
46	<<5 20 19 28 20 16 27 -12 -4 13]]		42.869	81/80 99/98 2541/2500
47	<<0 0 0 31 0 0 49 0 72 87]]		42.959	81/80 126/125 1029/1024
48	<<11 -18 -14 12 -54 -53 -19 18 90 82]]		44.005	225/224 441/440 9317/9216
49	<<11 13 17 43 -5 -4 30 3 55 62]]		44.041	126/125 176/175 2430/2401
50	<<13 21 6 17 3 -27 -18 -45 -33 27]]		44.186	121/120 441/440 891/875
51	<<26 11 12 34 -43 -54 -36 -3 41 54]]		44.660	176/175 1331/1323 2401/2400
52	<<61 -4 21 106 -148 -138 -43 60 260 225]]		44.729	3025/3024 5632/5625 825000/823543
53	<<60 -8 11 119 -152 -151 -19 48 304 296]]		44.888	2401/2400 46656/46585 172032/171875
54	<<19 -17 4 32 -71 -47 -15 57 133 76]]		45.168	225/224 385/384 585640/583443
55	<<5 -11 -12 28 -29 -33 27 3 103 120]]		45.456	225/224 385/384 27783/27500
56	<<5 20 19 -3 20 16 -22 -12 -76 -74]]		45.466	81/80 540/539 1375/1372
57	<<1 4 -21 -13 4 -36 -24 -60 -44 36]]		45.509	81/80 176/175 1331/1323
58	<<18 10 25 45 -26 -11 9 30 70 40]]		45.7950	176/175 243/242 1375/1372
59	<<39 1 18 82 -89 -81 -5 39 187 168]]		45.928	540/539 5632/5625 151263/151250
60	<<2 -23 -11 5 -41 -23 1 39 91 52]]		46.181	243/242 441/440 1815/1792
61	<<4 -15 -22 10 -33 -46 2 -9 75 104]]		46.462	225/224 243/242 1617/1600
62	<<9 5 28 7 -13 19 -20 51 -1 -77]]		47.885	121/120 225/224 891/875
63	<<33 8 20 67 -64 -61 -8 24 128 119]]		48.088	540/539 3136/3125 15488/15435
64	<<14 -6 -15 4 -42 -63 -42 -18 30 63]]		48.150	121/120 441/440 3136/3125
65	<<8 1 18 -11 -17 6 -45 39 -29 -93]]		48.186	225/224 385/384 891/875
66	<<50 -17 4 94 -143 -134 -24 57 277 250]]		48.798	2401/2400 3025/3024 1265625/1261568
67	<<7 -3 -23 2 -21 -56 -21 -45 15 85]]		49.572	121/120 126/125 2079/2048
68	<<20 18 14 19 -18 -34 -39 -18 -18 5]]		49.917	121/120 126/125 1728/1715

13-limit temperaments with badness below 0.04

Listed is the wedgie and the TE badness times 1000 for 64 temperaments with badness less than 0.04.

Rank	Wedgie	Name	Badness	Commas
1	<<6 -7 -2 15 -34 -25 -20 3 -76 15 59 -53 49 -88 -173]]	Benediction	15.715	225/224 243/242 351/350 441/440
2	<<10 9 7 25 -5 -9 -17 5 -45 -9 27 -45 46 -40 -110]]	Myna	17.125	126/125 144/143 176/175 196/195
3	<<1 4 10 18 15 4 13 25 20 12 28 20 16 5 -15]]	Meantone	18.048	66/65 81/80 99/98 105/104
4	<<6 -7 -2 15 -3 -25 -20 3 -27 15 59 19 49 -1 -66]]	Miraculous	18.669	105/104 144/143 196/195 275/273
5	<<7 -3 8 2 -19 -21 -7 -21 -56 27 15 -33 -22 -83 -73]]	Orwell	19.718	99/98 121/120 176/175 275/273
6	<<7 -3 8 2 12 -21 -7 -21 -7 27 15 39 -22 4 34]]	Winston	19.931	66/65 99/98 105/104 121/120
7	<<9 5 -3 7 -20 -13 -30 -20 -65 -21 -1 -65 30 -45 -95]]	Valentino	20.665	121/120 126/125 176/175 196/195
8	<<1 4 10 -13 15 4 13 -24 20 12 -44 20 -71 5 100]]	Meanpop	20.883	81/80 105/104 144/143 196/195
9	<<9 5 -3 7 11 -13 -30 -20 -16 -21 -1 7 30 42 12]]	Lupercalia	21.328	66/65 105/104 121/120 126/125
10	<<15 -2 -5 22 -23 -38 -50 -17 -92 -6 58 -46 79 -46 -161]]	Hemithirds	21.738	196/195 352/351 1001/1000 1029/1024
11	<<5 -11 -12 -3 -18 -29 -33 -22 -47 3 31 -1 33 -6 -51]]	Tritonic	22.993	105/104 121/120 196/195 275/273
12	<<16 2 5 40 -39 -34 -37 8 -121 6 86 -98 95 -128 -283]]	Hemiwürschmidt	23.074	243/242 351/350 441/440 3584/3575
13	<<11 13 17 12 10 -5 -4 -19 -25 3 -17 -25 -25 -35 -10]]	Nusecond	23.323	66/65 99/98 121/120 126/125
14	<<2 8 -11 5 -1 8 -23 1 -9 -48 -16 -32 52 38 -22]]	Mohajira	23.388	66/65 105/104 121/120 512/507
15	<<8 1 18 20 -4 -17 6 4 -36 39 43 -13 -6 -78 -88]]	Würschmidt	23.593	99/98 144/143 176/175 275/273
16	<<3 12 -1 -8 14 12 -10 -23 11 -36 -60 -12 -19 43 78]]	Mothra	23.954	81/80 99/98 105/104 144/143
17	<<4 16 9 10 29 16 3 2 31 -24 -32 8 -3 48 63]]	Agora	24.522	81/80 99/98 105/104 121/120
18	<<23 -1 13 42 -27 -55 -44 -13 -128 33 101 -59 73 -124 -249]]	Grendel	24.839	352/351 540/539 625/624 1375/1372
19	<<17 6 15 27 -24 -30 -24 -16 -101 18 42 -78 24 -123 -183]]		25.204	176/175 351/350 540/539 1375/1372
20	<<0 0 0 0 31 0 0 0 49 0 0 72 0 87 107]]	Gallium	25.484	81/80 99/98 121/120 126/125
21	<<4 16 9 10 -2 16 3 2 -18 -24 -32 -64 -3 -39 -44]]	Squares	25.514	66/65 81/80 99/98 121/120
22	<<1 4 10 18 -16 4 13 25 -29 12 28 -52 16 -82 -122]]	Grosstone	25.899	81/80 99/98 126/125 144/143
23	<<13 -10 6 17 -22 -46 -27 -18 -83 42 74 -14 27 -84 -139]]	Slender	25.913	225/224 275/273 385/384 1331/1323
24	<<21 -9 -7 37 -57 -63 -70 -14 -168 9 117 -99 128 -134 -334]]		27.405	385/384 441/440 625/624 13720/13689
25	<<2 8 20 5 30 8 26 1 40 24 -16 40 -55 10 85]]		27.835	81/80 105/104 121/120 196/195
26	<<2 8 20 5 -1 8 26 1 -9 24 -16 -32 -55 -77 -22]]	Migration	28.071	66/65 121/120 126/125 1215/1183
27	<<22 -5 3 24 -42 -59 -57 -38 -148 21 73 -79 57 -129 -234]]		28.384	352/351 385/384 625/624 1331/1323
28	<<17 6 15 27 7 -30 -24 -16 -52 18 42 -6 24 -36 -76]]		28.408	144/143 176/175 196/195 275/273
29	<<16 2 5 9 -8 -34 -37 -41 -72 6 14 -26 8 -41 -61]]		28.432	121/120 176/175 196/195 275/273
30	<<3 12 -1 23 14 12 -10 26 11 -36 12 -12 68 43 -37]]		29.207	66/65 81/80 105/104 1001/1000
31	<<6 -7 -2 -16 -3 -25 -20 -46 -27 15 -13 19 -38 -1 49]]		29.452	66/65 99/98 105/104 1001/1000
32	<<10 9 7 25 26 -9 -17 5 4 -9 27 27 46 47 -3]]		29.868	66/65 105/104 126/125 540/539
33	<<7 -3 8 33 -19 -21 -7 28 -56 27 87 -33 65 -83 -188]]		30.237	144/143 225/224 351/350 441/440
34	<<16 2 5 40 -8 -34 -37 8 -72 6 86 -26 95 -41 -176]]	Hemithir	31.199	144/143 196/195 243/242 625/624
35	<<32 4 10 49 -47 -68 -74 -33 -193 12 100 -124 103 -169 -344]]		31.732	352/351 1001/1000 1716/1715 3025/3024
36	<<10 9 7 -6 -5 -9 -17 -44 -45 -9 -45 -45 -41 -40 5]]		31.850	66/65 99/98 126/125 385/384
37	<<15 -2 -5 22 -54 -38 -50 -17 -141 -6 58 -118 79 -133 -268]]		31.990	351/350 385/384 441/440 3146/3125
38	<<23 -1 13 42 -58 -55 -44 -13 -177 33 101 -131 73 -211 -356]]		32.359	351/350 540/539 1375/1372 5632/5625
39	<<1 4 10 -13 -16 4 13 -24 -29 12 -44 -52 -71 -82 -7]]		33.007	66/65 81/80 126/125 385/384
40	<<12 -14 -4 30 -37 -50 -40 6 -103 30 118 -34 98 -89 -239]]		33.451	225/224 243/242 441/440 1875/1859
41	<<8 1 18 20 27 -17 6 4 13 39 43 59 -6 9 19]]		34.382	66/65 99/98 105/104 243/242
42	<<8 1 -13 20 -4 -17 -43 4 -36 -33 43 -13 101 37 -88]]		34.458	105/104 126/125 144/143 243/242
43	<<3 -19 -1 -8 -17 -37 -10 -23 -38 51 47 31 -19 -44 -29]]		35.530	99/98 105/104 121/120 640/637
44	<<8 1 -13 20 -35 -17 -43 4 -85 -33 43 -85 101 -50 -195]]		35.585	126/125 196/195 385/384 1575/1573
45	<<3 12 -1 -8 -17 12 -10 -23 -38 -36 -60 -84 -19 -44 -29]]		36.239	66/65 81/80 99/98 385/384
46	<<8 1 -13 -11 -4 -17 -43 -45 -36 -33 -29 -13 14 37 27]]		36.331	66/65 105/104 126/125 512/507
47	<<5 -11 -12 -3 -49 -29 -33 -22 -96 3 31 -73 33 -93 -158]]		36.533	121/120 225/224 351/350 441/440
48	<<10 9 7 -6 26 -9 -17 -44 4 -9 -45 27 -41 47 112]]		36.656	99/98 105/104 126/125 144/143
49	<<3 12 -1 23 -17 12 -10 26 -38 -36 12 -84 68 -44 -144]]		36.857	81/80 144/143 176/175 1029/1024
50	<<22 -5 3 55 -42 -59 -57 11 -148 21 145 -79 144 -129 -349]]		36.876	243/242 441/440 625/624 3584/3575
51	<<5 20 19 28 13 20 16 27 2 -12 -4 -44 13 -34 -59]]		37.382	66/65 81/80 99/98 1001/1000
52	<<4 -15 9 10 -2 -33 3 2 -18 63 75 51 -3 -39 -44]]		37.408	99/98 105/104 144/143 243/242
53	<<13 -10 6 17 -53 -46 -27 -18 -132 42 74 -86 27 -171 -246]]		37.732	225/224 351/350 385/384 1331/1323
54	<<12 17 27 30 25 -1 9 6 -5 15 11 -5 -9 -30 -25]]		37.849	66/65 99/98 126/125 243/242
55	<<0 0 0 31 0 0 0 49 0 0 72 0 87 0 -115]]		37.885	81/80 105/104 196/195 512/507
56	<<13 21 6 17 9 3 -27 -18 -34 -45 -33 -57 27 3 -32]]		38.116	66/65 121/120 343/338 441/440
57	<<10 9 7 25 -36 -9 -17 5 -94 -9 27 -117 46 -127 -217]]		38.811	126/125 176/175 243/242 1573/1568
58	<<7 -3 8 33 12 -21 -7 28 -7 27 87 39 65 4 -81]]		38.812	105/104 196/195 275/273 648/637
59	<<9 5 -3 38 -20 -13 -30 29 -65 -21 71 -65 117 -45 -210]]		39.221	126/125 144/143 196/195 1029/1024
60	<<25 7 2 47 -28 -47 -67 -12 -137 -15 85 -91 125 -86 -271]]		39.271	196/195 352/351 1001/1000 6144/6125
61	<<14 -6 16 35 -38 -42 -14 7 -112 54 102 -66 43 -166 -261]]		39.531	225/224 243/242 351/350 2420/2401
62	<<11 13 17 12 41 -5 -4 -19 24 3 -17 47 -25 52 97]]		39.576	99/98 105/104 121/120 126/125
63	<<11 13 17 43 10 -5 -4 30 -25 3 55 -25 62 -35 -125]]		39.707	126/125 176/175 196/195 648/637
64	<<16 2 5 9 -39 -34 -37 -41 -121 6 14 -98 8 -128 -168]]		39.907	121/120 176/175 351/350 1375/1372